Vol. 41, N 3, 2007, pp. 447–459 www.edpsciences.org/m2an
DOI: 10.1051/m2an:2007030
MATHEMATICAL FRAMEWORK FOR CURRENT DENSITY IMAGING DUE TO DISCHARGE OF ELECTRO-MUSCULAR DISRUPTION DEVICES
Jeehyun Lee
1, Jin Keun Seo
2and Eung Je Woo
3Abstract. Electro-muscular disruption (EMD) devices such as TASER M26 and X26 have been used as a less-than-lethal weapon. Such EMD devices shoot a pair of darts toward an intended target to generate an incapacitating electrical shock. In the use of the EMD device, there have been controversial questions about its safety and effectiveness. To address these questions, we need to investigate the distribution of the current densityJinside the target produced by the EMD device. One approach is to develop a computational model providing a quantitative and reliable analysis about the distribution ofJ. In this paper, we set up a mathematical model of a typical EMD shock, bearing in mind that we are aiming to compute the current density distribution inside the human body with a pair of inserted darts. The safety issue of TASER is directly related to the magnitude of|J|at the region of the darts where the current density J is highly concentrated. Hence, fine computation of J near the dart is essential. For such numerical simulations, serious computational difficulties are encountered in dealing with the darts having two different very sharp corners, tip of needle and tip of barb. The boundary of a small fishhook-shaped dart inside a large computational domain and the presence of corner singularities require a very fine mesh leading to a formidable amount of numerical computations. To circumvent these difficulties, we developed a multiple point source method of computing J. It has a potential to provide effective analysis and more accurate estimate of Jnear fishhook-shaped darts. Numerical experiments show that the MPSM is just fit for the study of EMD shocks.
Mathematics Subject Classification. 93A30, 32S05, 92C55, 33K28, 35M10, 35R30.
Received March 7, 2006.
1. Introduction
Electro-muscular disruption (EMD) devices such as TASER M26 and X26 are being used as a less-than-lethal weapon [27]. These devices generate a powerful electrical shock to completely override the central nervous system of a victim and directly control a large amount of skeletal muscles. It shoots a pair of fishhook-shaped darts attached to 15-feet wires toward an intended target. When the darts are inserted in the victims’s skin or clothing, the device generates a debilitating electrical shock. Although these devices are intended to reduce the probability of fatality or permanent injury compared with other lethal weapons, there are ongoing controversial
Keywords and phrases.Electro-muscular disruption (EMD) device, electrical current density, Maxwell equations, non-smooth boundary, elliptic partial differential equations, corner singularity.
1 Department of Mathematics, Yonsei University, Korea.[email protected]
2 Department of Mathematics, Yonsei University and National Institute for Mathematical Science, Korea. [email protected]
3 College of Electronics and Information, Kyung Hee University, Korea.[email protected]
c EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.esaim-m2an.org or http://dx.doi.org/10.1051/m2an:2007030
9mm 0.8mm
Figure 1. Fishhook-shaped dart used in typical EMD devices.
arguments over its safety as the number of death increases with more frequent uses of such EMD devices. See [3, 6, 11, 13, 14, 29]. Numerous investigations on the implications of EMD shocks are, therefore, requested to create a guidance that helps device users to minimize the risk of injury or fatality of victims.
When the pair of darts strike a victim and an electrical circuit is formed, a current densityJ= (Jx, Jy, Jz) is established inside the body. The distribution ofJis influenced by several factors such as the shape and location of darts, distance between darts, geometry of the body, and its conductivity distribution. The knowledge ofJ is necessary for the investigation of the risk of ventricular fibrilation (VF), seizure, skin burn, and so on.
In this paper, we suggest a basic study to undertake a three-dimensional numerical modelling of an EMD shock as a way to assess its biophysical implications. We setup a mathematical model of the EMD shock that can correctly simulate the real situation. Accurate computation ofJnear the dart is required to properly assess the risk of skin burn injury and VF in the worst case. In numerical computations ofJusing such a model, however, there are serious difficulties in dealing with the region of the fishhook-shaped dart sinceJis highly concentrated there. As shown in Figure 1, the dart is not axially symmetric and the inserted dart creates a three-dimensional computational domain with two different corners, tip of needle and tip of barb, where singularities of|J|occur.
Therefore, dealing with these corners requires a very fine mesh to get enough numerical accuracy, and it may lead to a formidable amount of numerical computations and memory space. To circumvent these difficulties, we develop a multiple point source method (MPSM) of computingJ, that provides an analytical representation formula for the potential distribution near the fishhook-shaped dart. Eliminating the requirement of a very fine mesh, the MPSM seems to be an ideal method for us to model EMD shocks using a conventional PC.
In this paper, we try to explain the whole process of the current density imaging during EMD shocks;
motivation, mathematical medelling, computational method, and a future study of experimental validation using animal subjects. We hope that the proposed computer model to simulate the effect of shocks help examine competing claims about the benefits and risks of TASERs [3, 13, 27].
2. Mathematical modelling of EMD shock 2.1. Problem definition
In this section, we formulate a mathematical model of EMD shocks, bearing in mind that we are aiming to compute the current density distribution inside the body generated by a pair of inserted darts. We denote the three-dimensional domain of the subject as Ω with its boundary∂Ω. Letr= (x, y, z) denote a position vector.
Since an EMD shock is induced by very short pulses of stimulating current, the subject Ω can be assumed to have an isotropic conductivityσthat is a positive function defined in Ω. The darts, denoted byE1andE2, are fired from the device and have a fishhook-shaped geometry with 9 mm length and 0.8 mm diameter as shown in Figure 1.
An EMD device transmits electricity to the body Ω through the pair of darts, and it stimulates the subject with about 2 to 15 A body current and 50 000 V peak voltage. We assume the injection current through the pair of darts isIA. The currentIgives rise to a current densityJ= (Jx, Jy, Jz) inside the subject Ωs:= Ω\ E1∪ E2
Figure 2. Simplified model of a computational domain with two darts.
depicted in Figure 2. For a given amount of injection currentI, we have the following data along the darts:
−
∂E1
J·nds=I=
∂E2
J·nds
where∂Ejis the boundary of the dartEj intersecting Ω,nthe unit outward normal vector to the boundary∂Ωs, and ds the surface area element. The current densityJcan be expressed as
J= (Jx, Jy, Jz) =−σ∇V (1)
where V is the voltage inside the body Ωs. Since the dart is a very good conductor, V can be assumed to be constant on each dart. Setting the voltage on the dartE2to be zero, the voltageV satisfies the following mixed boundary value problem with the elliptic PDE:
⎧⎪
⎪⎨
⎪⎪
⎩
∇ ·(σ(r)∇V(r)) = 0 in Ωs:= Ω\ E1∪ E2
V = a constant on∂E1, V = 0 on∂E2
σ∂V∂n = 0 on ∂Ω\ E1∪ E2
∂E1σ∂V∂n ds=I=−
∂E2σ∂V∂n ds
(2)
where ∂V∂n is the outward normal derivative ofV. Here,∂Ej is a Lipschitz boundary.
The model (2) is inconvenient as a computational model, so we need to modify it maintaining its structure.
For a moment, we assume the voltageV at the dartE1 isαand letu=2Vα −1.Thenusatisfies the following
PDE: ⎧
⎨
⎩
∇ ·(σ∇u) = 0 in Ωs
u= 1 on E1 and u=−1 on E2
∂n∂u = 0 on ∂Ω\ E1∪ E2.
(3) We need to determineαto computeV directly fromuusing the relationV = α2(u+1). Taking normal derivative to V = α2(u+ 1) and integrating it over∂E1, we have
∂E1σ∂V∂n ds = α2
∂E1σ∂u∂n ds and obtain the following linear relation:
V(r) =
∂E1σ∂u∂nds
I (u(r) + 1), r∈Ωs.
Hence, it suffices to computeuin the simplified model (3) for the computation ofJ=−σ∇V in the model (2).
From this observation, it is easy to see the well-posedness of the model (2).
For an accurate analysis of the distribution ofJnear the darts, we should carefully model the exact shape of the dart depicted in Figure 1. Assuming that the dartE1is lying along thez-axis, we may express its geometry as follows:
E1:=
r∈R3:
x2+y2<0.4, 0< z≤6
∪
r∈R3:
x2+ (y+152(z−6))2≤2(9−z)15 , 6≤z <9
∪
r∈R3:
x2+ (y+65(z−6.4))2≤0.2(z−5.4), 5.4≤z <6.4 .
(4)
2.2. Concentration of J near fishhook-shaped dart
In the region where darts are inserted, very high concentration ofJis established. To see this in a quantitative way, let us compare |J|on E1 with|J| over the zero equipotential surface Γ0:={r∈Ω : u(r) = 0}. We can view Γ0roughly as a cross-sectional surface lying somewhere between two darts since Γ0 separates the body Ωs into two parts Ω+s := {r∈Ωs : u(r)>0}and Ω−s = Ωs\Ω+s. According to the divergence theorem, we have
0 =
Ω+s
∇ ·(σ∇u)dr=
∂Ω+s
σ∂u
∂n dS=
∂E1
σ∂u
∂ndS+
Γ0
σ∂u
∂n dS
where Ω+s := {r∈Ωs : u(r)>0}. Since the normal vector on the equipotential surface is n= |∇u|∇u on∂E1
andn=|∇u|∇u on Γ0, the above identity becomes
∂E1
|J|dS=c
∂E1
|σ∇u|dS=c
Γ0
|σ∇u|dS=
Γ0
|J|dS where c = 1I
∂E1σ∂u∂nds is the scaling constant. Note that the surface area of the dart is approximately 0.8π×9 mm2and the surface area of a cross-section of a normal adult body would be bigger than 200×300 mm2. Hence, we may conclude that
[average|J| on∂E1] ≥ 2000× [average|J|on the cross-section Γ0].
For the risk characterization, therefore, we must focus onJnear the darts, while we may disregardJat some distance from the darts. In particular, we should carefully investigate the magnitude of J near the dart E1
where it is placed with the shortest distance to the heart as the worst case.
3. Multiple point source method (MPSM) 3.1. Homogeneous half-space model
In the model (3), we need to develop a computational method providing reliable estimates ofJ near darts with various positions. The computational domain has two corners at the dart, tip of needle and tip of barb, where singularities of J occur. Since the tip and barb of the dart of this TASER model do not have any radial symmetry, any asymptotic formula of the gradient of the solution in this TASER model is not available.
Conventional FEM requires a very fine mesh near the fishhook-shaped corners to get enough numerical accuracy and it leads to a large-scale computation. The proposed MPSM effectively deals with complicated singularities at very sharp small corners.
To explain the MPSM, we begin with considering the case where the subject is the upper half-spaceR3+= {r= (x, y, z) : z >0}and has a homogeneous conductivity. Let u1 be the solution of (3) with Ω replaced by
R3+ and limr→∞u1(r) = 0. Then we have
⎧⎪
⎪⎨
⎪⎪
⎩
∇2u1= 0 in R3+\ E1∪ E2
u1= 1 on ∂E1 and u=−1 on ∂E2
∂u1
∂n = 0 on ∂R3+\ E1∪ E2
limr→∞u1(r) = 0
(5)
where∇2=∇ · ∇denotes the Laplacian.
The basic idea of the MPSM is to view the dart as a sum of a sequence of point sources lying inside the dart and their image sources with respect to the z-plane. Leta0,a1, . . . ,aM1 be a sequence of points lying inside the dart E1 and leta−j be the reflected point of aj to the z-plane. Similarly, let b0, . . . ,bM2 ∈ E2 with their reflected pointsb−j . Now we consider the multiple-source function
W(r) =
M1
j=0
q1j
1
|r−aj| + 1
|r−a−j|
−
M2
j=0
qj2
1
|r−bj|+ 1
|r−b−j|
(6)
whereqj1andq2j are positive point charges. ThisW satisfies
∇2W(r) = 0 in Ωs=R3+\(E1∪ E2) and lim
|r|→∞W(r) = 0. Due to the presence of the image sources ata−j andb−j, we have
∂
∂zW(r) = 0 on thez-plane.
Hence, W satisfies all the conditions of the model (3) but the boundary conditions on the darts. If we could approximateW ≈1 onE1andW ≈ −1 onE2, thenW ≈u1in the entire domain Ωsaccording to the standard maximum principle.
In order for W to be a good approximation of u1, we need to find appropriate source points and point charges such thatW ≈1 onE1andW ≈ −1 onE2. To be precise, we need to find the vectorsa0, . . . ,aM1 ∈ E1, b0, . . . ,bM2∈ E2, and positive quantitiesq01, . . . , qM11, q02, . . . , q2M1 which minimize the following functional
∂E1|1−W(r)|2ds+
∂E2|1 +W(r)|2ds
=
∂E1
1−M1
j=0q1jFj1(r) + M2
j=0qj2Fj2(r)2 ds +
∂E2
1 +M1
j=0qj1Fj1(r) − M2
j=0qj2Fj2(r)2 ds
(7)
where
Fj1(r) := 1
|r−aj|+ 1
|r−a+j| and Fj2(r) := 1
|r−bj|+ 1
|r−b+j|·
To simplify the minimization problem, we may chooseaj andbj by considering the distribution of the electric potential due to the special geometry of the fishhook-shaped dartE1. For example, one can choose
aj =
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
0, −0.4 +25(1.4)j−8, 9−3(1.4)j−8
, 0≤j ≤7
0, 0, 5(j−8)8
, 8≤j ≤15
0, 0, 5 +j−168
, 16≤j≤23
0, 6(j−24)40 , 6.4−j−248
, 24≤j≤33.
Some theoretical insight about corner singularity is generally required for an appropriate choice of such source points.
With the given a0, . . . ,aM1,b0, . . . ,bM2, the minimization problem in (7) can be reduced to find a vector q= (q10, . . . , q1M1, q02, . . . , qM22) minimizing the following functional:
Φ(q) =
∂E1
|1−q·F(r)|2 ds+
∂E2
|1 +q·F(r)|2 ds (8)
where
F(r) =
F01(r), . . . , FM11(r),−F02(r), . . . ,−FM22(r) .
To discretize (8), we choose a sequence of points on the boundary of darts such as r11, . . . ,r1N
1 ∈ ∂E1 and r21, . . . ,r2N
2 ∈∂E2. Then, this minimization problem can be simplified to find qthat minimizes the following sum:
Φ(q) =˜
N1
k=0
1−q·F(r1k)2+
N2
k=0
1 +q·F(r2k)2. (9) For readers who try to test the MPSM, we provide an example ofr1j:
r1j =
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
0, −0.4, 9−3(1.4)j−N
, j= 0, . . . , N−1
0, 0.4, 5(j−N)N
, N ≤j≤2N−1
0, 0.4, 5 + (j−2NN )
, 2N≤j≤3N−1
0, 0.4 + 6(j−3N5N ), 6.4−j−3NN
3N≤j≤4N−1
0, 0.4, 6(j−4N)N
, 4N≤j≤5N−1
and N = 8.
Ifq∗ is a minimizer of the functional ˜Φ(q) in (9), then the solutionu1 can be approximated as
u1(r) ≈ q∗·F(r), r∈R3+\ E1∪ E2. (10)
3.1.1. Remark on MPSM
The MPSM is very flexible and it can be viewed as a dart-drawing process by appropriately arranging point sources at aj. The choice of the number of poles should depend on the global geometry of the dart and the singular structure of the gradient of the solution. Some knowledge about corner singularities in elliptic PDE and layer potential theory may help determine a desirable arrangement ofaj by looking at the geometry of the dart, or one can learn it heuristically by using a computer program. In particular, since the barb of the dart is very thin, we need to arrange poles densely, and position of them should be located a little bit on the upper part of the center line of the barb due to the influence of poles of the main cylindrical part of the dart from which the barb branches out.
Let us state a simple theoretical lemma that is related to the MPSM.
Lemma 3.1. Letu1be the solution of(5). There existq1∈L2(∂E1∗)andq2∈L2(∂E2∗)such thatq1andq2 are symmetric with respect toz-axis and
u1(r) =S∂E1∗q1(r)− S∂E2∗q2(r) inR3+\ E1∪ E2
where Ej∗ is the interior domain of Ej∪ {r∈R3 : (x, y,−z)∈ Ej}, j = 1,2 and S∂Ej∗qj is the single layer potentialS∂Ej∗qj(r) :=
∂Ej∗ −1
4π|r−r|qj(r) dsr.
(a) (b)
Figure 3. (a) Hexahedral domain and (b) human body model.
Proof. We will give a brief proof. We extend u1 into the lower half-space by even reflection, so the extended functionu∗1 is defined byu∗1(x, y,−z) =u1(x, y, z) for (x, y, z)∈R3+\ E1∪ E2. Thenu∗1 satisfies the Dirichlet problem
∇2u∗1= 0 in R3\ E1∗∪ E2∗, u∗1= 1 on∂E1∗, u∗1=−1 on∂E2∗.
Then the proof follows from the invertibility of single layer potential on Lipschitz domain [28].
Since the dart Ej is very thin and sharp, it is like a curve. Hence, the above lemma can be modified as follows. For appropriately given curvesγj contained inEj∗, there exist chargesqj ∈L2(γj) such that
γ1
1
4π|r−r|q1(r)dlr−
γ2
1
4π|r−r|q2(r)dlr ≈
1 on ∂E1
−1 on ∂E2
where dl is the length element. The MPSM can be viewed as a discretized version of the above approximation.
3.2. Computation of J in bounded domain and finite geometry effect
3.2.1. Hexahedral domain with homogeneous conductivityNow, we consider the solutionuof (3) in a bounded domain Ωswith a homogeneous conductivity. We begin with the case where Ω is a hexahedral domain withσ= 1 as shown in Figure 3a. To be precise, let
Ω :={r= (x, y, z) : −100 mm< x <400 mm, |y|<100 mm, 0< z <150 mm}. (11) Assume that the dart E1 strikes at the origin. The other dartE2 strikesL mm below E1 and 100 mm< L <
300 mm, that is,E2={r+ (L,0,0) : r∈ E1}.
For the computation ofu, we make a good use of the approximationu1(r) ≈ q∗·F(r) in (10). We decompose the solutionuof (3) into
u(r) =u1(r) +φ(r), r∈Ωs. Then,φsatisfies the following mixed boundary value problem:
⎧⎨
⎩
∇2φ= 0 in Ωs φ= 0 on ∂E1∪∂E2
∂φ∂n=−∂u∂n1 on ∂Ω\ E1∪ E2.
(12)
The quantity∇φcan be viewed as a finite geometry effect. Due to the presence of the sharp darts E1 andE2 in Ω, the computation of φis again a formidable job. In order to avoid this, we decompose φagain into two parts:
φ(r) =φ1(r) +φ2(r)
whereφ1 is the solution of the following standard Neumann boundary problem:
∇2φ1= 0 in Ω
n· ∇φ2=−n· ∇u1 on ∂Ω,
∂Ωφ1= 0 (13)
andφ2 satisfies ⎧
⎨
⎩
∇2φ2= 0 in Ωs
φ2=−φ1 on ∂E1∪∂E2 n· ∇φ2= 0 on ∂Ω\ E1∪ E2.
(14) Note that the computation ofφ1 is easyviaany standard numerical method without using a fine mesh.
Next, we consider the second termφ2. We conjecture that the ratio |∇φ|∇u|2| is very small in Ωs∩BwhereB is the ball with 30 mm radius and centered at the origin. Then we can neglect the second termφ2 in Ωs∩B, the region of interest. Although this conjecture seems to be obvious, we are not able to rigorously prove it. Indeed, even the clear fact that|∇u|>0 in Ωs∩B is very difficult to be proved in a three-dimensional case, while there are some related two-dimensional results in [1, 2, 23]. We think that these theoretical issues should be addressed in future studies, although those are obvious from physical insight.
Without a rigorous proof, we try to explain a reason why |∇φ|∇u|2| is very small in Ωs∩B. According to the expression ofu1 in (10), the quantity|∇u1(r)| decays with a rate of the square of the distance from the dart.
The solution φ1 of (14) has the Neumann boundary condition ∂φ∂n1 =−∂u∂n1 which is very small on ∂Ω due to the decaying property of u1. Hence, the contribution of ∇φ1 to ∇u near the dart would be very small. We performed numerical computations of φ1 with various 100 mm ≤ L ≤ 300 mm and the results showed that
|∇φ1| ≈0 nearE1 and the difference of φ1 betweenE1 andE2 are less than 0.01. These crude estimates yield that φ1 ≈c1 on E1 and φ1 ≈c2 on E2 and 0 <|c1−c2| <0.01. Since φ2 satisfies the same boundary value problem withuin (3) except some difference on∂E1∪∂E2,ucan be approximated by
u(r)≈ 2
c1−c2φ2(r) + 1− 2c1
c1−c2·
Hence, |∇φ|∇u|2|≈ |c1−c2 2|≤0.005 in the region Ωs∩B. We again stress that this estimate is not proven rigorously, so it is an open problem.
The above estimate means that φ2 has less than 1% effect on J in the region of interest. Hence, we can approximate
∇u≈ ∇u1+∇φ1≈ ∇(q∗·F(r)) +∇φ1. (15) The major advantage of this approximation is that the two terms of the right side of (15) are computable using a PC. Various numerical simulations show that u1+φ1 satisfies (3) with the Dirichlet conditionsu1+φ1≈1 onE1 andu1+φ1≈ −1 onE2. Hence,u1+φ1 can be viewed as an accurate numerical solution of (3).
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.4
Figure 4. Modelling of the fishhook-shaped dart using the MPSM. White dots inside the dart are source points.
3.2.2. Human body model with homogeneous conductivity
Now, let Ω be the human body model shown in Figure 3b and let Ω∗ be the hexahedral domain in (11).
The three-dimensional model in Figure 3b was generated by using the Visible Korean Human data set [20]. We still assume that the conductivityσ is homogeneous. When the origin is positioned at the pit of the stomach, most adult human bodies can contain the hexahedral domain Ω∗with an appropriate coordinate system. Then we are considering a worst case where the dartE1 strikes at the pit of the stomach. For simplicity, we assume Ω∗∩{z= 0} ⊂Ω∩{z= 0}, which is reasonable. Since∂Ω∩{z= 0}is outside the hexahedron Ω∗, the boundary effect to Jnear the dart is smaller than that is in the case of Ω∗ only. For the human body model Ω under the assumption of a homogeneous conductivity, the approximation (15) also holds.
4. Numerical experiments 4.1. Modelling of fishhook-shaped dart using MPSM
The MPSM provides an efficient way to model the fishhook-shaped dart by arranging point sources inside the dart. A sequence of source pointsaj anda−j were chosen as in 3.1 by considering the voltage distribution due to the special geometry of the dart. Now, the shape modelling problem reduces to find appropriate point charges to enforce the boundary condition on the dart. In order to solve this, we employed the discrete minimization using boundary points of the dart, ri given in 3.1. Then we approximated multiple-source function W and Figure 4 shows the contour plot of the obtained voltage distribution. Equipotential surface ofV = 1 denoted as black color matches well with the fishhook-shaped dart.
4.2. Distribution of voltage
Figure 5a–c show distributions of voltage from the homogeneous half-space model with increasing field-of- view. Figure 6a–c are the corresponding voltage distributions computed from the hexagonal model. Using the homogeneous human body model, voltage distributions were almost identical to those from the hexagonal model.
As expected, the voltage distribution near each dart in the bounded domain changes little compared with that in the half-space. In the region far away from both darts, we can observe differences in voltage distributions.
However, the gradients of voltage there are small.
-2 -1 0 1 2 0
1 2 3 4 5
0.6 0.7 0.8 0.9
-5 0 5
0 2 4 6 8 10
0.6 0.7 0.8 0.9
-2.5 2.5 -10 -5 0 5 10 15 20 25 30
0 5 10 15
0.2 0.4 0.6 0.8
(a) (b) (c)
Figure 5. Distribution of voltage from the half-space model with different field-of-view: (a) 5×
5 cm2, (b) 10×10 cm2and (c) 40×15 cm2.
-2 -1 0 1 2
0 1 2 3 4 5
0.6 0.7 0.8 0.9
-5 0 5
0 2 4 6 8 10
0.6 0.7 0.8 0.9
-2.5 2.5 -10 -5 0 5 10 15 20 25 30
0 5 10 15
0.2 0.4 0.6 0.8
(a) (b) (c)
Figure 6. Distribution of voltage from the hexagonal model with different field-of-view: (a) 5×
5 cm2, (b) 10×10 cm2and (c) 40×15 cm2.
4.3. Distribution of magnitude of current density
Figure 7a–c are distributions of magnitude of current density from the homogeneous half-space model with increasing field-of-view. Figure 8a–c are the corresponding distributions of magnitude of current density from the hexagonal model. The homogeneous human body model produced almost identical results as in Figure 8a–c.
Figure 9 shows how the magnitude of current density decays as we move along the straight line starting from the tip of the dart. In all three homogeneous models, the plots are almost identical within 13 cm distance from the tip of the dart.
5. Discussions on realistic human body model and animal experiments
The conductivity distribution of a realistic human body model should not be homogeneous since different tissues have different conductivity values. For a realistic human body model, therefore, we need a three- dimensional human anatomy data set with knowledge of conductivity distribution. As illustrated in Figure 3b, we are using the Visible Korean Human data set that includes segmented cross-sectional images of a Korean adult male cadaver [20]. We are currently conducting numerical simulations using this realistic human body model by assigning appropriate conductivity values to the segmented internal regions. Though this ongoing research requires a great deal of efforts and time, we speculate that the three-dimensional realistic human body model together with the MPSM suggested in this paper will be able to provide the most reliable numerical simulation results of EMD shocks. Especially we should focus on properly incorporating an inhomogeneous
-2 -1 0 1 2 0
1 2 3 4 5
10 30 50 70
-5 0 5
0 2 4 6 8 10
10 30 50 70
-2.5 2.5 -10 -5 0 5 10 15 20 25
0 5 10 15
5 10 15 20
5
(a) (b) (c)
Figure 7. Distribution of magnitude of current density from the half-space model with differ- ent field-of-view: (a) 5×5 cm2, (b) 10×10 cm2 and (c) 40×15 cm2.
-2 -1 0 1 2
0 1 2 3 4 5
10 30 50 70
-5 0 5
0 2 4 6 8 10
10 30 50 70
-2.5 2.5 -10 -5 0 5 10 15 20 25 30
0 5 10 15
5 10 15 20
(a) (b) (c)
Figure 8. Distribution of magnitude of current density from the hexagonal model with differ- ent field-of-view: (a) 5×5 cm2, (b) 10×10 cm2 and (c) 40×15 cm2.
13 13.2 13.4 13.6 13.8 14
4.5 5 5.5 6 x 10-4
Halfspace Bounded domain
13 13.2 13.4 13.6 13.8 14
4.5 5 5.5 6 x 10-4
Halfspace Bounded domain
0 2 4 6 8 10 12 14
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Halfspace Bounded domain
d=0
d
d
d
(a) (b) (c)
Figure 9. (a) We are interested in the magnitude of current density along the dotted line where d is the distance from the tip of the dart. (b) Decay of magnitude of current density wherex-axis is the distancedfrom the tip of the dart. (c) is the magnified plot of the portion circled in (b).
realistic conductivity distribution into the human body model since it is highly possible that such a conductivity distribution will alter voltage and current density distributions even in a region near the fishhook-shaped dart.
Using the realistic human body model, we apply the approximation (10) inside a small region of pixels containing the dart to handle singularities at the sharp corners. For example,
∇u(r)≈∇u1(r) +∇φ1(r) in Ωs∩(D1∪D2) (16) where D1={r : |x|< α, |y|< α, 0< z < α+ 10},D2 ={r+ (L,0,0) : r∈D1}, andαis a small number depending on a given pixel size of the segmented anatomy image data. For the computation ofuin Ω\D1∪D2, we may solve the following problem:
⎧⎨
⎩
∇ ·(σ∇u) = 0 in Ω\D1∪D2
σ∂∂un=σ∂u∂n1 +σ∂φ∂n1 on ∂(D1∪D2)∩Ω σ∂∂un= 0 on ∂Ω\D1∪D2.
(17)
Finally, let us discuss the assignment of conductivity values to different tissues inside the realistic human body model. Conductivity image reconstruction from a living human body has been a long-term goal in the area of electrical impedance tomography (EIT) whose corresponding mathematical problem is the inverse problem of identifying the elliptic coefficient from Dirichlet-to-Neumann map. In EIT, elegant mathematical theory and reconstruction algorithms have been developed [4, 5, 9, 15, 16, 21, 26]. However, in clinical environments, EIT has not reached yet to the stage providing reliable static conductivity values due to its ill-posed nature.
Recently, magnetic resonance electrical impedance tomography (MREIT) has been proposed to deal with the ill-posed nature of EIT [7, 8, 10, 12, 17–19, 22, 24, 25]. MREIT uses an MRI system to get a supplementary data, that is one componentBz of the magnetic flux densityB= (Bx, By, Bz), whereBis governed by Amp`ere law∇ ×B=−σ∇uanduis the solution of the elliptic PDE: ∇ ·(σ∇u) = 0 with an injected Neumann data (or current). Numerous numerical simulations and experiments have shown that MREIT can produce high- resolution conductivity images, provided that we inject enough amount of current (or L1-norm of Neumann data).
In our future experimental studies to validate numerical simulation results, we plan to first perform con- ductivity imaging of animal subjects using MREIT techniques. Once we obtain high-resolution conductivity images, then we can generate current density images by solving the direct problem with various position of darts using the method described in this paper. This kind of experimental validations of our numerical method using animal experiments will be invaluable for the interpretation of any numerical results using a realistic human body model since experimental studies using living human subjects are not possible.
Acknowledgements. This work was supported by the SRC/ERC program of MOST/KOSEF (R11-2002-103 and R01- 2005-000-10339-0). Much of this work was done during the visit of IMA, University of Minnesota. We would like to thank IMA. We would like to thank Professor John G. Webster at the Department of Biomedical Engineering, University of Wisconsin-Madison for introducing us this interesting problem and providing valuable technical information on EMD devices. We would like to also thank Professor Min Suk Chung at the Department of Anatomy, Aju University School of Medicine, Korea for providing us the Visible Korean Human data set.
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